use each digit (from the given base set) just once to produce the best possible approximation of Pi

[Joshua Zucker]

On Tue, Mar 25, 2008 at 6:06 AM, Alexander Povolotsky
<apovolot at gmail.com> wrote:
> Greetings to All,
>
>  I noticed that
>
>  689725314 / 219546387 = 76636146 / 24394043 = 3.141592642...
>
>  Of course the *quality of approximation" is very poor (16 digits "in"
>  are yielding just 8 accurate digits) but please note that in the ratio
>  below
>
>  6897253140 / 2195463870
>
>  all 10 digits of the decimal base (from 0 to 9) are used just once in
>  both numerator and denominator.
>  I hope that I found the *best* combination of digits for both
>  numerator and denominator (but I am notsure since I've done it late
>  night by hand).
>
>  I wonder if anyone would be interested to see what other (than decimal
>  ) base systems could give with regards to the same approach of using
>  each digit (from the given base set) just once to produce the best
>  possible approximation of Pi.
>

Assuming that rounding error of double precision (16 digits) is good
enough, the best fraction using all 10 digits in numerator and
denominator is
8405139762 / 2675439081
3.141592653591...

To answer the question about other bases,
base 4: 3210/1023 (base 4) = 3.04 (base 10) -- best we can do?  I think so.
base 5: 42301/12043 (base 5) = 3.14699 (base 10).
base 6: 431502/124053 = 3.14158899
base 7: 3251406/1036245 = 3.14159208
base 8: 50316247/14670523 = 3.14159266311
base 9: 743865021/234175068 = 3.141592653514

I don't think this is interesting enough to submit, but if anyone
cares to, let me know (and I suppose we'll want the base 10 versions
of these numerators and denominators?).

--Joshua Zucker